Differentials of One Variable

    • assignment Zapping With d 1

      assignment Homework

      Zapping With d 1
      Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

      Find the differential of each of the following expressions; zap each of the following with \(d\):

      1. \[f=3x-5z^2+2xy\]

      2. \[g=\frac{c^{1/2}b}{a^2}\]

      3. \[h=\sin^2(\omega t)\]

      4. \[j=a^x\]

      5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

    • assignment Approximating a Delta Function with Isoceles Triangles

      assignment Homework

      Approximating a Delta Function with Isoceles Triangles
      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]

      Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].

      1. Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
      2. Using the test function \(f(x)=3x^2\), find the value of \[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\] Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).

    • assignment Flux through a Plane

      assignment Homework

      Flux through a Plane
      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
    • assignment The puddle

      assignment Homework

      The puddle
      differentials AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
      1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
      2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
      3. FOOD FOR THOUGHT (optional)
        There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
    • assignment Find Area/Volume from $d\vec{r}$

      assignment Homework

      Find Area/Volume from \(d\vec{r}\)
      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021

      Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

      1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
      2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
      3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

    • assignment Rubber Sheet

      assignment Homework

      Rubber Sheet
      Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

      Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call \(y\), and the distance of horizontal stretch we will call \(x\).

      If I pull the bottom down by a small distance \(\Delta y\), with no horizontal force, what is the resulting change in width \(\Delta x\)? Express your answer in terms of partial derivatives of the potential energy \(U(x,y)\).

    • format_list_numbered Integration Sequence

      format_list_numbered Sequence

      Integration Sequence
      Students learn/review how to do integrals in a multivariable context, using the vector differential \(d\vec{r}=dx\, \hat{x}+dy\, \hat{y}+dz\, \hat{z}\) and its curvilinear coordinate analogues as a unifying strategy. This strategy is common among physicists, but is NOT typically taught in vector calculus courses and will be new to most students.
    • assignment Cube Charge

      assignment Homework

      Cube Charge
      charge density

      Integration Sequence

      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022
      1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
      2. In a new physical situation: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.
    • assignment_ind Vector Differential--Rectangular

      assignment_ind Small White Board Question

      10 min.

      Vector Differential--Rectangular
      Vector Calculus II Fall 2021 AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Vector Calculus II Summer 21 Static Fields Winter 2021

      vector differential rectangular coordinates math

      Integration Sequence

      In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

      This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

    • group Scalar Surface and Volume Elements

      group Small Group Activity

      30 min.

      Scalar Surface and Volume Elements
      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022 Static Fields Winter 2022

      Integration Sequence

      Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

      This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

  • AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022 Find the total differential of the following functions:
    1. \(y=3x^2 + 4\cos 2x\)
    2. \(y=3x^2\cos kx\) (where \(k\) is a constant)
    3. \(y=\frac{\cos 7x}{x^2}\)
    4. \(y=\cos(3x^2-2)\)